A Hamiltonian cycle (path) of a graph G is a simple cycle (path) which contains all the vertices of G. The Hamiltonian cycle problem asks whether a given graph contains a Hamiltonian cycle. It is NP-complete even for 3-connected planar graphs [3, 61. However, the problem becomes polynomial-time solvable for Cconnected planar graphs: Tutte proved that such a graph necessarily contains a Hamilton...

Definition 1 A Hamiltonian cycle in a graph is a cycle that visits each vertex exactly once. A Hamiltonian graph is a graph that contains a Hamiltonian cycle. It is well known that the problem of determining if a graph is Hamiltonian is N P-complete. Here we will construct a NIZK proof in the hidden-bits model (HBM) that is able to prove that a graph is Hamiltonian. First we define how graphs a...

The generalised Sudoku problem with N symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where N is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, w...

The balanced Hamiltonian cycle problem is a quiet new topic of graph theorem. Given a graph G = (V, E), whose edge set can be partitioned into k dimensions, for positive integer k and a Hamiltonian cycle C on G. The set of all i-dimensional edge of C, which is a subset by E(C), is denoted as Ei(C). If ||Ei(C)| |Ej(C)|| 1 for 1 i < j k, C is called a balanced Hamiltonian cycle. In this paper, th...

Let G be a 2-connected weighted graph such that the minimum weighted degree is at least d. In [1], Bondy and Fan proved that either G contains a cycle of weight at least 2d or every heaviest cycle in G is a hamiltonian cycle. If G is not hamiltonian, this theorem implies the existence of a cycle of weight at least 2d, but in case of G is hamiltonian we cannot decide whether G has a heavy cycle ...

The traveling salesman problem (TSP) is one of the most fundamental problems in combinatorial optimization. Given a graph, the goal is to find a Hamiltonian cycle of minimum or maximum weight. We consider finding Hamiltonian cycles of maximum weight (Max-TSP). An instance of Max-TSP is a complete graph G = (V,E) with edge weights w : E → N. The goal is to find a Hamiltonian cycle of maximum wei...

In this paper, we discuss hamiltonian problems for reducible Powgraphs. The main result is finding, in linear time, the unique hamiltonian cycle, if it exists. In order to obtain this result, two other related problems are solved: finding the hamiltonian path starting at the source vertex and finding the hamiltonian cycle given the hamiltonian path.